url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | linarith | m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : m ≠ 0
⊢ n ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : m ≠ 0
⊢ n ≠ 0
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | define | case h.mpr.left
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ ↑a ≡ ↑b (MOD m) ∧ ↑a ≡ ↑c (MOD n)
⊢ a ∈ Set_rp_below (m * n) | case h.mpr.left
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ ↑a ≡ ↑b (MOD m) ∧ ↑a ≡ ↑c (MOD n)
⊢ rel_prime (m * n) a ∧ a < m * n | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.left
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ ↑a ≡ ↑b (MOD m) ∧ ↑a ≡ ↑c (MOD n)
⊢ a ∈ Set_rp_below (m * n)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | apply And.intro _ h5.left | case h.mpr.left
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ ↑a ≡ ↑b (MOD m) ∧ ↑a ≡ ↑c (MOD n)
⊢ rel_prime (m * n) a ∧ a < m * n | m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ ↑a ≡ ↑b (MOD m) ∧ ↑a ≡ ↑c (MOD n)
⊢ rel_prime (m * n) a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.left
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ ↑a ≡ ↑b (MOD m) ∧ ↑a ≡ ↑c (MOD n)
⊢ rel_prime (m * n) a ∧ a < m * n
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | rewrite [Lemma_7_4_6] | m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ ↑a ≡ ↑b (MOD m) ∧ ↑a ≡ ↑c (MOD n)
⊢ rel_prime (m * n) a | m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ ↑a ≡ ↑b (MOD m) ∧ ↑a ≡ ↑c (MOD n)
⊢ rel_prime m a ∧ rel_prime n a | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ ↑a ≡ ↑b (MOD m) ∧ ↑a ≡ ↑c (MOD n)
⊢ rel_prime (m * n) a
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | rewrite [congr_rel_prime h5.right.left,
congr_rel_prime h5.right.right] | m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ ↑a ≡ ↑b (MOD m) ∧ ↑a ≡ ↑c (MOD n)
⊢ rel_prime m a ∧ rel_prime n a | m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ ↑a ≡ ↑b (MOD m) ∧ ↑a ≡ ↑c (MOD n)
⊢ rel_prime m b ∧ rel_prime n c | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ ↑a ≡ ↑b (MOD m) ∧ ↑a ≡ ↑c (MOD n)
⊢ rel_prime m a ∧ rel_prime n a
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | show rel_prime m b ∧ rel_prime n c from
And.intro h2.left.left h2.right.left | m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ ↑a ≡ ↑b (MOD m) ∧ ↑a ≡ ↑c (MOD n)
⊢ rel_prime m b ∧ rel_prime n c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ ↑a ≡ ↑b (MOD m) ∧ ↑a ≡ ↑c (MOD n)
⊢ rel_prime m b ∧ rel_prime n c
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | rewrite [congr_iff_mod_eq_Nat, congr_iff_mod_eq_Nat] at h5 | case h.mpr.right
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ ↑a ≡ ↑b (MOD m) ∧ ↑a ≡ ↑c (MOD n)
⊢ mod_mod m n a = (b, c) | case h.mpr.right
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ a % m = b % m ∧ a % n = c % n
⊢ mod_mod m n a = (b, c) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.right
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ ↑a ≡ ↑b (MOD m) ∧ ↑a ≡ ↑c (MOD n)
⊢ mod_mod m n a = (b, c)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | rewrite [mod_mod_def, h5.right.left, h5.right.right] | case h.mpr.right
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ a % m = b % m ∧ a % n = c % n
⊢ mod_mod m n a = (b, c) | case h.mpr.right
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ a % m = b % m ∧ a % n = c % n
⊢ (b % m, c % n) = (b, c) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.right
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ a % m = b % m ∧ a % n = c % n
⊢ mod_mod m n a = (b, c)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | rewrite [Nat.mod_eq_of_lt h2.left.right,
Nat.mod_eq_of_lt h2.right.right] | case h.mpr.right
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ a % m = b % m ∧ a % n = c % n
⊢ (b % m, c % n) = (b, c) | case h.mpr.right
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ a % m = b % m ∧ a % n = c % n
⊢ (b, c) = (b, c) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.right
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ a % m = b % m ∧ a % n = c % n
⊢ (b % m, c % n) = (b, c)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | rfl | case h.mpr.right
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ a % m = b % m ∧ a % n = c % n
⊢ (b, c) = (b, c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.right
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ a % m = b % m ∧ a % n = c % n
⊢ (b, c) = (b, c)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.eq_numElts_of_equinum | [1188, 1] | [1193, 7] | rewrite [numElts_def] at h2 | U V : Type
A : Set U
B : Set V
n : ℕ
h1 : A ∼ B
h2 : numElts A n
⊢ numElts B n | U V : Type
A : Set U
B : Set V
n : ℕ
h1 : A ∼ B
h2 : I n ∼ A
⊢ numElts B n | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A : Set U
B : Set V
n : ℕ
h1 : A ∼ B
h2 : numElts A n
⊢ numElts B n
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.eq_numElts_of_equinum | [1188, 1] | [1193, 7] | rewrite [numElts_def] | U V : Type
A : Set U
B : Set V
n : ℕ
h1 : A ∼ B
h2 : I n ∼ A
⊢ numElts B n | U V : Type
A : Set U
B : Set V
n : ℕ
h1 : A ∼ B
h2 : I n ∼ A
⊢ I n ∼ B | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A : Set U
B : Set V
n : ℕ
h1 : A ∼ B
h2 : I n ∼ A
⊢ numElts B n
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.eq_numElts_of_equinum | [1188, 1] | [1193, 7] | show I n ∼ B from Theorem_8_1_3_3 h2 h1 | U V : Type
A : Set U
B : Set V
n : ℕ
h1 : A ∼ B
h2 : I n ∼ A
⊢ I n ∼ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A : Set U
B : Set V
n : ℕ
h1 : A ∼ B
h2 : I n ∼ A
⊢ I n ∼ B
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Theorem_7_4_4 | [1195, 1] | [1206, 7] | have h2 : numElts (Set_rp_below m) (phi m) := phi_is_numElts m | m n : ℕ
h1 : rel_prime m n
⊢ phi (m * n) = phi m * phi n | m n : ℕ
h1 : rel_prime m n
h2 : numElts (Set_rp_below m) (phi m)
⊢ phi (m * n) = phi m * phi n | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
⊢ phi (m * n) = phi m * phi n
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Theorem_7_4_4 | [1195, 1] | [1206, 7] | have h3 : numElts (Set_rp_below n) (phi n) := phi_is_numElts n | m n : ℕ
h1 : rel_prime m n
h2 : numElts (Set_rp_below m) (phi m)
⊢ phi (m * n) = phi m * phi n | m n : ℕ
h1 : rel_prime m n
h2 : numElts (Set_rp_below m) (phi m)
h3 : numElts (Set_rp_below n) (phi n)
⊢ phi (m * n) = phi m * phi n | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
h2 : numElts (Set_rp_below m) (phi m)
⊢ phi (m * n) = phi m * phi n
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Theorem_7_4_4 | [1195, 1] | [1206, 7] | have h4 : numElts (Set_rp_below (m * n)) (phi (m * n)) :=
phi_is_numElts (m * n) | m n : ℕ
h1 : rel_prime m n
h2 : numElts (Set_rp_below m) (phi m)
h3 : numElts (Set_rp_below n) (phi n)
⊢ phi (m * n) = phi m * phi n | m n : ℕ
h1 : rel_prime m n
h2 : numElts (Set_rp_below m) (phi m)
h3 : numElts (Set_rp_below n) (phi n)
h4 : numElts (Set_rp_below (m * n)) (phi (m * n))
⊢ phi (m * n) = phi m * phi n | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
h2 : numElts (Set_rp_below m) (phi m)
h3 : numElts (Set_rp_below n) (phi n)
⊢ phi (m * n) = phi m * phi n
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Theorem_7_4_4 | [1195, 1] | [1206, 7] | have h5 : numElts (Set_rp_below m ×ₛ Set_rp_below n) (phi (m * n)) :=
eq_numElts_of_equinum (Set_rp_below_prod h1) h4 | m n : ℕ
h1 : rel_prime m n
h2 : numElts (Set_rp_below m) (phi m)
h3 : numElts (Set_rp_below n) (phi n)
h4 : numElts (Set_rp_below (m * n)) (phi (m * n))
⊢ phi (m * n) = phi m * phi n | m n : ℕ
h1 : rel_prime m n
h2 : numElts (Set_rp_below m) (phi m)
h3 : numElts (Set_rp_below n) (phi n)
h4 : numElts (Set_rp_below (m * n)) (phi (m * n))
h5 : numElts (Set_rp_below m ×ₛ Set_rp_below n) (phi (m * n))
⊢ phi (m * n) = phi m * phi n | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
h2 : numElts (Set_rp_below m) (phi m)
h3 : numElts (Set_rp_below n) (phi n)
h4 : numElts (Set_rp_below (m * n)) (phi (m * n))
⊢ phi (m * n) = phi m * phi n
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Theorem_7_4_4 | [1195, 1] | [1206, 7] | have h6 : numElts (Set_rp_below m ×ₛ Set_rp_below n) (phi m * phi n) :=
numElts_prod h2 h3 | m n : ℕ
h1 : rel_prime m n
h2 : numElts (Set_rp_below m) (phi m)
h3 : numElts (Set_rp_below n) (phi n)
h4 : numElts (Set_rp_below (m * n)) (phi (m * n))
h5 : numElts (Set_rp_below m ×ₛ Set_rp_below n) (phi (m * n))
⊢ phi (m * n) = phi m * phi n | m n : ℕ
h1 : rel_prime m n
h2 : numElts (Set_rp_below m) (phi m)
h3 : numElts (Set_rp_below n) (phi n)
h4 : numElts (Set_rp_below (m * n)) (phi (m * n))
h5 : numElts (Set_rp_below m ×ₛ Set_rp_below n) (phi (m * n))
h6 : numElts (Set_rp_below m ×ₛ Set_rp_below n) (phi m * phi n)
⊢ phi (m * n) = phi m * phi n | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
h2 : numElts (Set_rp_below m) (phi m)
h3 : numElts (Set_rp_below n) (phi n)
h4 : numElts (Set_rp_below (m * n)) (phi (m * n))
h5 : numElts (Set_rp_below m ×ₛ Set_rp_below n) (phi (m * n))
⊢ phi (m * n) = phi m * phi n
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Theorem_7_4_4 | [1195, 1] | [1206, 7] | show phi (m * n) = phi m * phi n from numElts_unique h5 h6 | m n : ℕ
h1 : rel_prime m n
h2 : numElts (Set_rp_below m) (phi m)
h3 : numElts (Set_rp_below n) (phi n)
h4 : numElts (Set_rp_below (m * n)) (phi (m * n))
h5 : numElts (Set_rp_below m ×ₛ Set_rp_below n) (phi (m * n))
h6 : numElts (Set_rp_below m ×ₛ Set_rp_below n) (phi m * phi n)
⊢ phi (m * n) = phi m * phi n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
h2 : numElts (Set_rp_below m) (phi m)
h3 : numElts (Set_rp_below n) (phi n)
h4 : numElts (Set_rp_below (m * n)) (phi (m * n))
h5 : numElts (Set_rp_below m ×ₛ Set_rp_below n) (phi (m * n))
h6 : numElts (Set_rp_below m ×ₛ Set_rp_below... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Theorem_8_2_1_1 | [1213, 1] | [1233, 7] | rewrite [ctble_iff_equinum_set_nat] at h1 | U V : Type
A : Set U
B : Set V
h1 : ctble A
h2 : ctble B
⊢ ctble (A ×ₛ B) | U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ctble B
⊢ ctble (A ×ₛ B) | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A : Set U
B : Set V
h1 : ctble A
h2 : ctble B
⊢ ctble (A ×ₛ B)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Theorem_8_2_1_1 | [1213, 1] | [1233, 7] | rewrite [ctble_iff_equinum_set_nat] at h2 | U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ctble B
⊢ ctble (A ×ₛ B) | U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
⊢ ctble (A ×ₛ B) | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ctble B
⊢ ctble (A ×ₛ B)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Theorem_8_2_1_1 | [1213, 1] | [1233, 7] | obtain (I : Set Nat) (h3 : I ∼ A) from h1 | U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
⊢ ctble (A ×ₛ B) | U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
⊢ ctble (A ×ₛ B) | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
⊢ ctble (A ×ₛ B)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Theorem_8_2_1_1 | [1213, 1] | [1233, 7] | obtain (J : Set Nat) (h4 : J ∼ B) from h2 | U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
⊢ ctble (A ×ₛ B) | U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
⊢ ctble (A ×ₛ B) | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
⊢ ctble (A ×ₛ B)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Theorem_8_2_1_1 | [1213, 1] | [1233, 7] | have h5 : I ×ₛ J ∼ A ×ₛ B := Theorem_8_1_2_1 h3 h4 | U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
⊢ ctble (A ×ₛ B) | U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
⊢ ctble (A ×ₛ B) | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
⊢ ctble (A ×ₛ B)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Theorem_8_2_1_1 | [1213, 1] | [1233, 7] | have h6 : I ×ₛ J ⊆ Univ (Nat × Nat) := by
fix p : Nat × Nat
assume h6 : p ∈ I ×ₛ J
show p ∈ Univ (Nat × Nat) from elt_Univ p
done | U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
⊢ ctble (A ×ₛ B) | U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
h6 : I ×ₛ J ⊆ Univ (ℕ × ℕ)
⊢ ctble (A ×ₛ B) | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
⊢ ctble (A ×ₛ B)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Theorem_8_2_1_1 | [1213, 1] | [1233, 7] | have h7 : ctble (Univ (Nat × Nat)) := by
define apply Or.inr
rewrite [denum_def]
show Univ Nat ∼ Univ (Nat × Nat) from Theorem_8_1_3_2 NxN_equinum_N
done | U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
h6 : I ×ₛ J ⊆ Univ (ℕ × ℕ)
⊢ ctble (A ×ₛ B) | U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
h6 : I ×ₛ J ⊆ Univ (ℕ × ℕ)
h7 : ctble (Univ (ℕ × ℕ))
⊢ ctble (A ×ₛ B) | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
h6 : I ×ₛ J ⊆ Univ (ℕ × ℕ)
⊢ ctble (A ×ₛ B)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Theorem_8_2_1_1 | [1213, 1] | [1233, 7] | have h8 : ctble (I ×ₛ J) := Exercise_8_1_17 h6 h7 | U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
h6 : I ×ₛ J ⊆ Univ (ℕ × ℕ)
h7 : ctble (Univ (ℕ × ℕ))
⊢ ctble (A ×ₛ B) | U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
h6 : I ×ₛ J ⊆ Univ (ℕ × ℕ)
h7 : ctble (Univ (ℕ × ℕ))
h8 : ctble (I ×ₛ J)
⊢ ctble (A ×ₛ B) | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
h6 : I ×ₛ J ⊆ Univ (ℕ × ℕ)
h7 : ctble (Univ (ℕ × ℕ))
⊢ ctble (A ×ₛ B)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Theorem_8_2_1_1 | [1213, 1] | [1233, 7] | show ctble (A ×ₛ B) from ctble_of_equinum_ctble h5 h8 | U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
h6 : I ×ₛ J ⊆ Univ (ℕ × ℕ)
h7 : ctble (Univ (ℕ × ℕ))
h8 : ctble (I ×ₛ J)
⊢ ctble (A ×ₛ B) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
h6 : I ×ₛ J ⊆ Univ (ℕ × ℕ)
h7 : ctble (Univ (ℕ × ℕ))
h8 : ctble (I ×ₛ J)
⊢ ctble (A ×ₛ B)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Theorem_8_2_1_1 | [1213, 1] | [1233, 7] | fix p : Nat × Nat | U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
⊢ I ×ₛ J ⊆ Univ (ℕ × ℕ) | U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
p : ℕ × ℕ
⊢ p ∈ I ×ₛ J → p ∈ Univ (ℕ × ℕ) | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
⊢ I ×ₛ J ⊆ Univ (ℕ × ℕ)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Theorem_8_2_1_1 | [1213, 1] | [1233, 7] | assume h6 : p ∈ I ×ₛ J | U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
p : ℕ × ℕ
⊢ p ∈ I ×ₛ J → p ∈ Univ (ℕ × ℕ) | U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
p : ℕ × ℕ
h6 : p ∈ I ×ₛ J
⊢ p ∈ Univ (ℕ × ℕ) | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
p : ℕ × ℕ
⊢ p ∈ I ×ₛ J → p ∈ Univ (ℕ × ℕ)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Theorem_8_2_1_1 | [1213, 1] | [1233, 7] | show p ∈ Univ (Nat × Nat) from elt_Univ p | U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
p : ℕ × ℕ
h6 : p ∈ I ×ₛ J
⊢ p ∈ Univ (ℕ × ℕ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
p : ℕ × ℕ
h6 : p ∈ I ×ₛ J
⊢ p ∈ Univ (ℕ × ℕ)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Theorem_8_2_1_1 | [1213, 1] | [1233, 7] | define | U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
h6 : I ×ₛ J ⊆ Univ (ℕ × ℕ)
⊢ ctble (Univ (ℕ × ℕ)) | U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
h6 : I ×ₛ J ⊆ Univ (ℕ × ℕ)
⊢ finite (Univ (ℕ × ℕ)) ∨ denum (Univ (ℕ × ℕ)) | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
h6 : I ×ₛ J ⊆ Univ (ℕ × ℕ)
⊢ ctble (Univ (ℕ × ℕ))
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Theorem_8_2_1_1 | [1213, 1] | [1233, 7] | apply Or.inr | U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
h6 : I ×ₛ J ⊆ Univ (ℕ × ℕ)
⊢ finite (Univ (ℕ × ℕ)) ∨ denum (Univ (ℕ × ℕ)) | case h
U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
h6 : I ×ₛ J ⊆ Univ (ℕ × ℕ)
⊢ denum (Univ (ℕ × ℕ)) | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
h6 : I ×ₛ J ⊆ Univ (ℕ × ℕ)
⊢ finite (Univ (ℕ × ℕ)) ∨ denum (Univ (ℕ × ℕ))
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Theorem_8_2_1_1 | [1213, 1] | [1233, 7] | rewrite [denum_def] | case h
U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
h6 : I ×ₛ J ⊆ Univ (ℕ × ℕ)
⊢ denum (Univ (ℕ × ℕ)) | case h
U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
h6 : I ×ₛ J ⊆ Univ (ℕ × ℕ)
⊢ Univ ℕ ∼ Univ (ℕ × ℕ) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
h6 : I ×ₛ J ⊆ Univ (ℕ × ℕ)
⊢ denum (Univ (ℕ × ℕ))
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Theorem_8_2_1_1 | [1213, 1] | [1233, 7] | show Univ Nat ∼ Univ (Nat × Nat) from Theorem_8_1_3_2 NxN_equinum_N | case h
U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
h6 : I ×ₛ J ⊆ Univ (ℕ × ℕ)
⊢ Univ ℕ ∼ Univ (ℕ × ℕ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
U V : Type
A : Set U
B : Set V
h1 : ∃ I, I ∼ A
h2 : ∃ I, I ∼ B
I : Set ℕ
h3 : I ∼ A
J : Set ℕ
h4 : J ∼ B
h5 : I ×ₛ J ∼ A ×ₛ B
h6 : I ×ₛ J ⊆ Univ (ℕ × ℕ)
⊢ Univ ℕ ∼ Univ (ℕ × ℕ)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | rewrite [Theorem_8_1_5_2] at h1 | U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ctble F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
⊢ ctble (⋃₀ F) | U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
⊢ ctble (⋃₀ F) | Please generate a tactic in lean4 to solve the state.
STATE:
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ctble F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
⊢ ctble (⋃₀ F)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | rewrite [Theorem_8_1_5_2] | U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
⊢ ctble (⋃₀ F) | U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
⊢ ∃ R, fcnl_onto_from_nat R (⋃₀ F) | Please generate a tactic in lean4 to solve the state.
STATE:
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
⊢ ctble (⋃₀ F)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | obtain (R : Rel Nat (Set U)) (h3 : fcnl_onto_from_nat R F) from h1 | U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
⊢ ∃ R, fcnl_onto_from_nat R (⋃₀ F) | U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : fcnl_onto_from_nat R F
⊢ ∃ R, fcnl_onto_from_nat R (⋃₀ F) | Please generate a tactic in lean4 to solve the state.
STATE:
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
⊢ ∃ R, fcnl_onto_from_nat R (⋃₀ F)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | define at h3 | U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : fcnl_onto_from_nat R F
⊢ ∃ R, fcnl_onto_from_nat R (⋃₀ F) | U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
⊢ ∃ R, fcnl_onto_from_nat R (⋃₀ F) | Please generate a tactic in lean4 to solve the state.
STATE:
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : fcnl_onto_from_nat R F
⊢ ∃ R, fcnl_onto_from_nat R (⋃₀ F)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | have Runiqueval : unique_val_on_N R := h3.left | U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
⊢ ∃ R, fcnl_onto_from_nat R (⋃₀ F) | U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
⊢ ∃ R, fcnl_onto_from_nat R (⋃₀ F) | Please generate a tactic in lean4 to solve the state.
STATE:
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
⊢ ∃ R, fcnl_onto_from_nat R (⋃₀ F)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | have Ronto : nat_rel_onto R F := h3.right | U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
⊢ ∃ R, fcnl_onto_from_nat R (⋃₀ F) | U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
⊢ ∃ R, fcnl_onto_from_nat R (⋃₀ F) | Please generate a tactic in lean4 to solve the state.
STATE:
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
⊢ ∃ R, fcnl_onto_from_nat R (⋃₀ F)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | set S : Rel Nat U := enum_union_fam F f R | U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
⊢ ∃ R, fcnl_onto_from_nat R (⋃₀ F) | U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
⊢ ∃ R, fcnl_onto_from_nat R (⋃₀ F) | Please generate a tactic in lean4 to solve the state.
STATE:
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
⊢ ∃ R, fcnl_onto_from_na... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | apply Exists.intro S | U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
⊢ ∃ R, fcnl_onto_from_nat R (⋃₀ F) | U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
⊢ fcnl_onto_from_nat S (⋃₀ F) | Please generate a tactic in lean4 to solve the state.
STATE:
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_unio... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | define | U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
⊢ fcnl_onto_from_nat S (⋃₀ F) | U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
⊢ unique_val_on_N S ∧ nat_rel_onto S (⋃₀ F) | Please generate a tactic in lean4 to solve the state.
STATE:
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_unio... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | apply And.intro | U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
⊢ unique_val_on_N S ∧ nat_rel_onto S (⋃₀ F) | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
⊢ unique_val_on_N S
case right
U : Typ... | Please generate a tactic in lean4 to solve the state.
STATE:
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_unio... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | define | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
⊢ unique_val_on_N S | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
⊢ ∀ ⦃n : ℕ⦄ ⦃x1 x2 : U⦄, S n x1 → S n x... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | fix n : Nat | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
⊢ ∀ ⦃n : ℕ⦄ ⦃x1 x2 : U⦄, S n x1 → S n x... | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
⊢ ∀ ⦃x1 x2 : U⦄, S n x1 → S n x2 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | fix a1 : U | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
⊢ ∀ ⦃x1 x2 : U⦄, S n x1 → S n x2 ... | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 : U
⊢ ∀ ⦃x2 : U⦄, S n a1 → S n... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | fix a2 : U | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 : U
⊢ ∀ ⦃x2 : U⦄, S n a1 → S n... | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 a2 : U
⊢ S n a1 → S n a2 → a1 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | assume Sna1 : S n a1 | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 a2 : U
⊢ S n a1 → S n a2 → a1 ... | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 a2 : U
Sna1 : S n a1
⊢ S n a2 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | assume Sna2 : S n a2 | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 a2 : U
Sna1 : S n a1
⊢ S n a2 ... | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 a2 : U
Sna1 : S n a1
Sna2 : S ... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | define at Sna1 | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 a2 : U
Sna1 : S n a1
Sna2 : S ... | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 a2 : U
Sna1 : ∃ p, fnnn p = n ... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | define at Sna2 | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 a2 : U
Sna1 : ∃ p, fnnn p = n ... | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 a2 : U
Sna1 : ∃ p, fnnn p = n ... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | obtain ((i1, j1) : Nat × Nat) (h4 : fnnn (i1, j1) = n ∧
∃ A ∈ F, R i1 A ∧ f A j1 a1) from Sna1 | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 a2 : U
Sna1 : ∃ p, fnnn p = n ... | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 a2 : U
Sna1 : ∃ p, fnnn p = n ... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | obtain (A1 : Set U) (Aija1 : A1 ∈ F ∧ R i1 A1 ∧ f A1 j1 a1)
from h4.right | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 a2 : U
Sna1 : ∃ p, fnnn p = n ... | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 a2 : U
Sna1 : ∃ p, fnnn p = n ... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | obtain ((i2, j2) : Nat × Nat) (h5 : fnnn (i2, j2) = n ∧
∃ A ∈ F, R i2 A ∧ f A j2 a2) from Sna2 | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 a2 : U
Sna1 : ∃ p, fnnn p = n ... | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 a2 : U
Sna1 : ∃ p, fnnn p = n ... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | obtain (A2 : Set U) (Aija2 : A2 ∈ F ∧ R i2 A2 ∧ f A2 j2 a2)
from h5.right | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 a2 : U
Sna1 : ∃ p, fnnn p = n ... | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 a2 : U
Sna1 : ∃ p, fnnn p = n ... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | rewrite [←h5.left] at h4 | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 a2 : U
Sna1 : ∃ p, fnnn p = n ... | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 a2 : U
Sna1 : ∃ p, fnnn p = n ... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | have h6 : (i1, j1) = (i2, j2) :=
fnnn_one_one (i1, j1) (i2, j2) h4.left | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 a2 : U
Sna1 : ∃ p, fnnn p = n ... | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 a2 : U
Sna1 : ∃ p, fnnn p = n ... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | have h7 : i1 = i2 ∧ j1 = j2 := Prod.mk.inj h6 | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 a2 : U
Sna1 : ∃ p, fnnn p = n ... | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 a2 : U
Sna1 : ∃ p, fnnn p = n ... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | rewrite [h7.left, h7.right] at Aija1 | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 a2 : U
Sna1 : ∃ p, fnnn p = n ... | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 a2 : U
Sna1 : ∃ p, fnnn p = n ... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | define at Runiqueval | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ℕ
a1 a2 : U
Sna1 : ∃ p, fnnn p = n ... | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : ∀ ⦃n : ℕ⦄ ⦃x1 x2 : Set U⦄, R n x1 → R n x2 → x1 = x2
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | have h8 : A1 = A2 := Runiqueval Aija1.right.left Aija2.right.left | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : ∀ ⦃n : ℕ⦄ ⦃x1 x2 : Set U⦄, R n x1 → R n x2 → x1 = x2
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ... | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : ∀ ⦃n : ℕ⦄ ⦃x1 x2 : Set U⦄, R n x1 → R n x2 → x1 = x2
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : ∀ ⦃n : ℕ⦄ ⦃x1 x2 : Set U⦄, R n x1 → R n x2 → x1 = x2
Ront... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | rewrite [h8] at Aija1 | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : ∀ ⦃n : ℕ⦄ ⦃x1 x2 : Set U⦄, R n x1 → R n x2 → x1 = x2
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ... | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : ∀ ⦃n : ℕ⦄ ⦃x1 x2 : Set U⦄, R n x1 → R n x2 → x1 = x2
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : ∀ ⦃n : ℕ⦄ ⦃x1 x2 : Set U⦄, R n x1 → R n x2 → x1 = x2
Ront... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | have fA2fcnlonto : fcnl_onto_from_nat (f A2) A2 := h2 A2 Aija2.left | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : ∀ ⦃n : ℕ⦄ ⦃x1 x2 : Set U⦄, R n x1 → R n x2 → x1 = x2
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ... | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : ∀ ⦃n : ℕ⦄ ⦃x1 x2 : Set U⦄, R n x1 → R n x2 → x1 = x2
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : ∀ ⦃n : ℕ⦄ ⦃x1 x2 : Set U⦄, R n x1 → R n x2 → x1 = x2
Ront... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | define at fA2fcnlonto | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : ∀ ⦃n : ℕ⦄ ⦃x1 x2 : Set U⦄, R n x1 → R n x2 → x1 = x2
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ... | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : ∀ ⦃n : ℕ⦄ ⦃x1 x2 : Set U⦄, R n x1 → R n x2 → x1 = x2
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : ∀ ⦃n : ℕ⦄ ⦃x1 x2 : Set U⦄, R n x1 → R n x2 → x1 = x2
Ront... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | have fA2uniqueval : unique_val_on_N (f A2) := fA2fcnlonto.left | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : ∀ ⦃n : ℕ⦄ ⦃x1 x2 : Set U⦄, R n x1 → R n x2 → x1 = x2
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ... | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : ∀ ⦃n : ℕ⦄ ⦃x1 x2 : Set U⦄, R n x1 → R n x2 → x1 = x2
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : ∀ ⦃n : ℕ⦄ ⦃x1 x2 : Set U⦄, R n x1 → R n x2 → x1 = x2
Ront... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | define at fA2uniqueval | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : ∀ ⦃n : ℕ⦄ ⦃x1 x2 : Set U⦄, R n x1 → R n x2 → x1 = x2
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ... | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : ∀ ⦃n : ℕ⦄ ⦃x1 x2 : Set U⦄, R n x1 → R n x2 → x1 = x2
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : ∀ ⦃n : ℕ⦄ ⦃x1 x2 : Set U⦄, R n x1 → R n x2 → x1 = x2
Ront... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | show a1 = a2 from fA2uniqueval Aija1.right.right Aija2.right.right | case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : ∀ ⦃n : ℕ⦄ ⦃x1 x2 : Set U⦄, R n x1 → R n x2 → x1 = x2
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
n : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : ∀ ⦃n : ℕ⦄ ⦃x1 x2 : Set U⦄, R n x1 → R n x2 → x1 = x2
Ront... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | define | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
⊢ nat_rel_onto S (⋃₀ F) | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
⊢ ∀ ⦃x : U⦄, x ∈ ⋃₀ F → ∃ n, S n x | Please generate a tactic in lean4 to solve the state.
STATE:
case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U :... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | fix x : U | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
⊢ ∀ ⦃x : U⦄, x ∈ ⋃₀ F → ∃ n, S n x | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
x : U
⊢ x ∈ ⋃₀ F → ∃ n, S n x | Please generate a tactic in lean4 to solve the state.
STATE:
case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U :... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | assume h4 : x ∈ ⋃₀ F | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
x : U
⊢ x ∈ ⋃₀ F → ∃ n, S n x | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : x ∈ ⋃₀ F
⊢ ∃ n, S n x | Please generate a tactic in lean4 to solve the state.
STATE:
case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U :... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | define at h4 | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : x ∈ ⋃₀ F
⊢ ∃ n, S n x | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ∈ F, x ∈ t
⊢ ∃ n, S n x | Please generate a tactic in lean4 to solve the state.
STATE:
case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U :... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | obtain (A : Set U) (h5 : A ∈ F ∧ x ∈ A) from h4 | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ∈ F, x ∈ t
⊢ ∃ n, S n x | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ∈ F, x ∈ t
A : Set U
h5... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U :... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | define at Ronto | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ∈ F, x ∈ t
A : Set U
h5... | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R n x
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ∈ F, x... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : nat_rel_onto R F
S : Rel ℕ U :... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | obtain (i : Nat) (h6 : R i A) from Ronto h5.left | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R n x
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ∈ F, x... | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R n x
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ∈ F, x... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | have fAfcnlonto : fcnl_onto_from_nat (f A) A := h2 A h5.left | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R n x
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ∈ F, x... | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R n x
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ∈ F, x... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | define at fAfcnlonto | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R n x
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ∈ F, x... | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R n x
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ∈ F, x... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | have fAonto : nat_rel_onto (f A) A := fAfcnlonto.right | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R n x
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ∈ F, x... | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R n x
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ∈ F, x... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | define at fAonto | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R n x
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ∈ F, x... | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R n x
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ∈ F, x... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | obtain (j : Nat) (h7 : f A j x) from fAonto h5.right | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R n x
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ∈ F, x... | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R n x
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ∈ F, x... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | apply Exists.intro (fnnn (i, j)) | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R n x
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ∈ F, x... | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R n x
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ∈ F, x... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | define | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R n x
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ∈ F, x... | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R n x
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ∈ F, x... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | apply Exists.intro (i, j) | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R n x
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ∈ F, x... | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R n x
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ∈ F, x... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | apply And.intro | case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R n x
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ∈ F, x... | case right.left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R n x
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | rfl | case right.left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R n x
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ∈... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.left
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | apply Exists.intro A | case right.right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R n x
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ... | case right.right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R n x
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_1 | [1235, 1] | [1304, 7] | show A ∈ F ∧ R (i, j).1 A ∧ f A (i, j).2 x from
And.intro h5.left (And.intro h6 h7) | case right.right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃ n, R n x
S : Rel ℕ U := enum_union_fam F f R
x : U
h4 : ∃ t ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.right
U : Type
F : Set (Set U)
f : Set U → Rel ℕ U
h1 : ∃ R, fcnl_onto_from_nat R F
h2 : ∀ A ∈ F, fcnl_onto_from_nat (f A) A
R : Rel ℕ (Set U)
h3 : unique_val_on_N R ∧ nat_rel_onto R F
Runiqueval : unique_val_on_N R
Ronto : ∀ ⦃x : Set U⦄, x ∈ F → ∃... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_2 | [1306, 1] | [1330, 7] | set f : Set U → Rel Nat U := fun (A : Set U) => Classical.choose (h1 A) | U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
h1 : ∀ (A : Set U), ∃ SA, A ∈ F → fcnl_onto_from_nat SA A
⊢ ∃ f, ∀ A ∈ F, fcnl_onto_from_nat (f A) A | U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
h1 : ∀ (A : Set U), ∃ SA, A ∈ F → fcnl_onto_from_nat SA A
f : Set U → Rel ℕ U := fun A => Classical.choose (_ : ∃ SA, A ∈ F → fcnl_onto_from_nat SA A)
⊢ ∃ f, ∀ A ∈ F, fcnl_onto_from_nat (f A) A | Please generate a tactic in lean4 to solve the state.
STATE:
U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
h1 : ∀ (A : Set U), ∃ SA, A ∈ F → fcnl_onto_from_nat SA A
⊢ ∃ f, ∀ A ∈ F, fcnl_onto_from_nat (f A) A
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_2 | [1306, 1] | [1330, 7] | apply Exists.intro f | U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
h1 : ∀ (A : Set U), ∃ SA, A ∈ F → fcnl_onto_from_nat SA A
f : Set U → Rel ℕ U := fun A => Classical.choose (_ : ∃ SA, A ∈ F → fcnl_onto_from_nat SA A)
⊢ ∃ f, ∀ A ∈ F, fcnl_onto_from_nat (f A) A | U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
h1 : ∀ (A : Set U), ∃ SA, A ∈ F → fcnl_onto_from_nat SA A
f : Set U → Rel ℕ U := fun A => Classical.choose (_ : ∃ SA, A ∈ F → fcnl_onto_from_nat SA A)
⊢ ∀ A ∈ F, fcnl_onto_from_nat (f A) A | Please generate a tactic in lean4 to solve the state.
STATE:
U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
h1 : ∀ (A : Set U), ∃ SA, A ∈ F → fcnl_onto_from_nat SA A
f : Set U → Rel ℕ U := fun A => Classical.choose (_ : ∃ SA, A ∈ F → fcnl_onto_from_nat SA A)
⊢ ∃ f, ∀ A ∈ F, fcnl_onto_from_nat (f A) A
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_2 | [1306, 1] | [1330, 7] | fix A : Set U | U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
h1 : ∀ (A : Set U), ∃ SA, A ∈ F → fcnl_onto_from_nat SA A
f : Set U → Rel ℕ U := fun A => Classical.choose (_ : ∃ SA, A ∈ F → fcnl_onto_from_nat SA A)
⊢ ∀ A ∈ F, fcnl_onto_from_nat (f A) A | U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
h1 : ∀ (A : Set U), ∃ SA, A ∈ F → fcnl_onto_from_nat SA A
f : Set U → Rel ℕ U := fun A => Classical.choose (_ : ∃ SA, A ∈ F → fcnl_onto_from_nat SA A)
A : Set U
⊢ A ∈ F → fcnl_onto_from_nat (f A) A | Please generate a tactic in lean4 to solve the state.
STATE:
U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
h1 : ∀ (A : Set U), ∃ SA, A ∈ F → fcnl_onto_from_nat SA A
f : Set U → Rel ℕ U := fun A => Classical.choose (_ : ∃ SA, A ∈ F → fcnl_onto_from_nat SA A)
⊢ ∀ A ∈ F, fcnl_onto_from_nat (f A) A
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_2 | [1306, 1] | [1330, 7] | show A ∈ F → fcnl_onto_from_nat (f A) A from Classical.choose_spec (h1 A) | U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
h1 : ∀ (A : Set U), ∃ SA, A ∈ F → fcnl_onto_from_nat SA A
f : Set U → Rel ℕ U := fun A => Classical.choose (_ : ∃ SA, A ∈ F → fcnl_onto_from_nat SA A)
A : Set U
⊢ A ∈ F → fcnl_onto_from_nat (f A) A | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
h1 : ∀ (A : Set U), ∃ SA, A ∈ F → fcnl_onto_from_nat SA A
f : Set U → Rel ℕ U := fun A => Classical.choose (_ : ∃ SA, A ∈ F → fcnl_onto_from_nat SA A)
A : Set U
⊢ A ∈ F → fcnl_onto_from_nat (f A) A
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_2 | [1306, 1] | [1330, 7] | fix A : Set U | U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
⊢ ∀ (A : Set U), ∃ SA, A ∈ F → fcnl_onto_from_nat SA A | U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
A : Set U
⊢ ∃ SA, A ∈ F → fcnl_onto_from_nat SA A | Please generate a tactic in lean4 to solve the state.
STATE:
U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
⊢ ∀ (A : Set U), ∃ SA, A ∈ F → fcnl_onto_from_nat SA A
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_2 | [1306, 1] | [1330, 7] | by_cases h2 : A ∈ F | U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
A : Set U
⊢ ∃ SA, A ∈ F → fcnl_onto_from_nat SA A | case pos
U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
A : Set U
h2 : A ∈ F
⊢ ∃ SA, A ∈ F → fcnl_onto_from_nat SA A
case neg
U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
A : Set U
h2 : A ∉ F
⊢ ∃ SA, A ∈ F → fcnl_onto_from_nat SA A | Please generate a tactic in lean4 to solve the state.
STATE:
U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
A : Set U
⊢ ∃ SA, A ∈ F → fcnl_onto_from_nat SA A
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_2 | [1306, 1] | [1330, 7] | have h3 : ctble A := h A h2 | case pos
U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
A : Set U
h2 : A ∈ F
⊢ ∃ SA, A ∈ F → fcnl_onto_from_nat SA A | case pos
U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
A : Set U
h2 : A ∈ F
h3 : ctble A
⊢ ∃ SA, A ∈ F → fcnl_onto_from_nat SA A | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
A : Set U
h2 : A ∈ F
⊢ ∃ SA, A ∈ F → fcnl_onto_from_nat SA A
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_2 | [1306, 1] | [1330, 7] | rewrite [Theorem_8_1_5_2] at h3 | case pos
U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
A : Set U
h2 : A ∈ F
h3 : ctble A
⊢ ∃ SA, A ∈ F → fcnl_onto_from_nat SA A | case pos
U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
A : Set U
h2 : A ∈ F
h3 : ∃ R, fcnl_onto_from_nat R A
⊢ ∃ SA, A ∈ F → fcnl_onto_from_nat SA A | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
A : Set U
h2 : A ∈ F
h3 : ctble A
⊢ ∃ SA, A ∈ F → fcnl_onto_from_nat SA A
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_2 | [1306, 1] | [1330, 7] | obtain (SA : Rel Nat U) (h4 : fcnl_onto_from_nat SA A) from h3 | case pos
U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
A : Set U
h2 : A ∈ F
h3 : ∃ R, fcnl_onto_from_nat R A
⊢ ∃ SA, A ∈ F → fcnl_onto_from_nat SA A | case pos
U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
A : Set U
h2 : A ∈ F
h3 : ∃ R, fcnl_onto_from_nat R A
SA : Rel ℕ U
h4 : fcnl_onto_from_nat SA A
⊢ ∃ SA, A ∈ F → fcnl_onto_from_nat SA A | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
A : Set U
h2 : A ∈ F
h3 : ∃ R, fcnl_onto_from_nat R A
⊢ ∃ SA, A ∈ F → fcnl_onto_from_nat SA A
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_2 | [1306, 1] | [1330, 7] | apply Exists.intro SA | case pos
U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
A : Set U
h2 : A ∈ F
h3 : ∃ R, fcnl_onto_from_nat R A
SA : Rel ℕ U
h4 : fcnl_onto_from_nat SA A
⊢ ∃ SA, A ∈ F → fcnl_onto_from_nat SA A | case pos
U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
A : Set U
h2 : A ∈ F
h3 : ∃ R, fcnl_onto_from_nat R A
SA : Rel ℕ U
h4 : fcnl_onto_from_nat SA A
⊢ A ∈ F → fcnl_onto_from_nat SA A | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
A : Set U
h2 : A ∈ F
h3 : ∃ R, fcnl_onto_from_nat R A
SA : Rel ℕ U
h4 : fcnl_onto_from_nat SA A
⊢ ∃ SA, A ∈ F → fcnl_onto_from_nat SA A
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_2 | [1306, 1] | [1330, 7] | assume h5 : A ∈ F | case pos
U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
A : Set U
h2 : A ∈ F
h3 : ∃ R, fcnl_onto_from_nat R A
SA : Rel ℕ U
h4 : fcnl_onto_from_nat SA A
⊢ A ∈ F → fcnl_onto_from_nat SA A | case pos
U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
A : Set U
h2 : A ∈ F
h3 : ∃ R, fcnl_onto_from_nat R A
SA : Rel ℕ U
h4 : fcnl_onto_from_nat SA A
h5 : A ∈ F
⊢ fcnl_onto_from_nat SA A | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
A : Set U
h2 : A ∈ F
h3 : ∃ R, fcnl_onto_from_nat R A
SA : Rel ℕ U
h4 : fcnl_onto_from_nat SA A
⊢ A ∈ F → fcnl_onto_from_nat SA A
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_8_2_2_2 | [1306, 1] | [1330, 7] | show fcnl_onto_from_nat SA A from h4 | case pos
U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
A : Set U
h2 : A ∈ F
h3 : ∃ R, fcnl_onto_from_nat R A
SA : Rel ℕ U
h4 : fcnl_onto_from_nat SA A
h5 : A ∈ F
⊢ fcnl_onto_from_nat SA A | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
U : Type
F : Set (Set U)
h : ∀ A ∈ F, ctble A
A : Set U
h2 : A ∈ F
h3 : ∃ R, fcnl_onto_from_nat R A
SA : Rel ℕ U
h4 : fcnl_onto_from_nat SA A
h5 : A ∈ F
⊢ fcnl_onto_from_nat SA A
TACTIC:
|
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